6.1 The Exponential Function and its Inverse

Hey guys,

So we are starting a new unit! This year is not going to be much different then what you have been learning in Grade 11.  Now, you will learn this later on but as an introduction - the main difference between what you will learn this year and last year is this:

Grade 11 Exponential Functions:

Solving for x when you have two exponential functions with the same base

ie.

27=3^x
3³=3^x (general form of: a^x=a^x)

Since we now have the same base, we can now only deal with the exponents.

3=x

or

y=b^x we are able to solve for x

Grade 12 Exponential Functions:

What if we have to solve for a^x = b^x ? This year, we will be learning how to solve for eponential equations that have different bases.

or in this unit since we will be dealing with inverse functions;

x=b^y is the inverse of y=b^x but how are we going to solve for y in the inverse exponential equation?

Just something for you guys to think about : )


Now for the summary of what we learned today in class.  Since we are just starting a new unit, we are familiar with most of the calculations already. 

First we know that exponential functions can be represented in an

Equation form:  The general form (or "naked" form as burchat described) is: y=b^x

ie: y =(2/3)^x y=3^x

Table of Values: Now the main feature you need to remember about the table of values is that your ratio of 1st differences = the base value

Last but not least,

Graph: By only looking at a graph representation, we are not 100% sure whether it is exponential and therefore need more information.  ie. a table of values to calculate the ratio of 1st differences.

Calculating the Rate of Change for Exponential Functions

Now calculating the IROC (instantaneous rate of change) for the exponential graph is just the same as we have done before in the past. 

1) Pick an x-value that is closest to the point you are going to do the IROC for
2) Make a table consisting of the chosen x-values and their corresponding y-values (you do this by substituting your chosen x-value(s) back into the original equation; your answer will be your y-value)
3) Using the information from the table you have created, you will calculate the two slopes of two secants that will be used to determine the instantaneous rate of change.  You will then calculate the AROC (average rate of change).  Select two points (the x-value and the corresponding y-value), alike what you learned in Grade 9: y₂-y₁ all over x₂-x₁ Remember communication is important!! your form should look like this:

 Remember to choose at least 2 points because the closer the secants approach the point we are interested in, the more accurate our IROC will be. 
4) Finally, choose the AROC that is the closest secant to the point we are interested in, this will be your IROC.  This secant will be the result of the closest pair calculation.  For example, if you are interested in the point x=0.  Your closest x-value would most likely either be 1.999 or 0.001 (this is just an example, you can always do more decimal places if you wish to be more accurate but after selecting 3 decimal places, you should be on the right track). 

Recall your inverse functions!

For this unit we will not only be looking at the exponential function but we will also be looking at it's inverse function. 

Some very very very important points you need to remember:

1) The graph of the inverse function is a reflection of the graph of the function in the line y=x.
2) To find the equation of the inverse function: switch the x&y and then solve for y.
3) The mapping rule that relates to the table of values of the inverse is given by simply switching the x&y values: (x,y) --> (y,x)

For the time being we will call the inverse of our (y=bx)exponential function, f^-1x. We will learn the proper form/term for this in tomorrow's lesson (we will then learn the beauty of log!)

A couple points that are helpful for you in this chapter will be:

1) The Rate of Change for the exponential function and their inverse function is the same! (as proven in the worksheet activity today)
2) In the exponential function, there will be a horizontal asymptote and in the inverse function, there will be a vertical asymptote
3) The transformations for these functions are very much similar to the way we have been dealing with transformations in the past (vertical stretch/compression, horizontal stretch/compression, vertical and horizontal translations)
4) TIP! Pay very close attention to your transformations because these can affect your asymptotes! For example, a vertical translation up or down applied on the exponential graph can alter your horizontal asymptote. Also pay very close attention to your y-intercept, this can also change depending on your transformations.

I believe that is about it! Good luck!